Alphabeta MathDouble-Affine Hecke Algebras and Macdonald Theory
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Rings, Subrings, Integral Domains and Fields
- Tensor Products of Modules
Categorification and Geometric Representation Theory
Real Analysis
2 · Logical flowchart
3 · Items
Definition
The rank-one root datum and parameter conventions for DAHA
Lemma
The rank-one DAHA Bernstein quotients are Laurent polynomials
Lemma
Distinct rank-one q-affine substitutions are linearly independent
Definition
The equal-parameter double-affine Hecke algebra of type A1
Lemma
Rank-one DAHA affine normal words span
Lemma
The rank-one DAHA difference-reflection operators satisfy the presentation
Definition
The rank-one DAHA polynomial representation
Theorem
The generic rank-one DAHA polynomial representation is faithful
Definition
The symmetric Macdonald difference operator of type A1
Remark
Skein and physical knot interfaces require separate DAHA maps
Lemma
The A1 Macdonald operator is triangular on Weyl orbit sums
Lemma
The symmetric DAHA translation restricts to the A1 Macdonald operator
Theorem